A364527 Triangle read by rows giving the number of square arrays composed of the numbers from 1 to n^2, counted up to rotation and reflection, with heterogeneity k, i.e., number of k different sums of rows, columns or diagonals with 1 <= k <= 2*n+2 for n > 1.

1, 0, 0, 0, 0, 3, 0, 1, 22, 346, 2060, 7989, 17160, 14662, 3120, 880

Offset: 1

Author

Martin Renner, Jul 27 2023

Keywords

• nonn
• more
• tabf

Comments

T(n,1) gives the number of magic squares A006052(n).

For n > 1, T(n,2*n+2) gives the number of squares with maximum heterogeneity, i.e., all sums are different (but do not necessarily form a sequence of consecutive integers), sometimes called (super)heterogeneous squares or antimagic squares.

Subsets of T(n,2) or T(n,3) with one or both of the diagonal sums not equal to the magic constant are sometimes called semimagic squares.

Sum_{k=1..2*n+2} T(n,k) = A086829(n) = (n^2)!/8 for n > 1.

Examples

			T(n,k) starts with
  n = 1: 1;
  n = 2: 0, 0, 0, 0, 3, 0;
  n = 3: 1, 22, 346, 2060, 7989, 17160, 14662, 3120;
etc.
For n = 2 there are only three square arrays up to rotation and reflection, all of heterogeneity k = 5, i.e.,
  [1 2] [1 2] [1 3]
  [3 4] [4 3] [4 2]
since there are always the five different sums of rows, columns and diagonals 3, 4, 5, 6 and 7.
For n = 3 the lexicographically first square arrays of heterogeneity 1 <= k <= 8 are
  [2 7 6] [1 2 6] [1 2 5] [1 2 3] [1 2 3] [1 2 3] [1 2 3] [1 2 3]
  [9 5 1] [5 9 4] [3 9 6] [5 6 4] [4 5 6] [4 5 7] [4 5 6] [4 5 8]
  [4 3 8] [3 7 8] [4 7 8] [9 7 8] [7 8 9] [6 9 8] [7 9 8] [6 9 7]
For k = 1 we have the famous Lo Shu square with magic sum (n^3+n)/2 = 15. The other sums for the given examples are (9, 18), (8, 18, 19), (6, 15, 18, 24), (6, 12, 15, 18, 24), (6, 11, 14 16, 18, 23), (6, 12, 14, 15, 16, 17, 24) and (6, 11, 13, 14, 16, 17, 18, 22). Note that there are different sets of sums, namely a total of 6 with two values, 61 with three, 348 with four, 1295 with five, 2880 with six, 3845 with seven and 1538 with eight.
		

References

• Pierre Berloquin, Garten der Sphinx. 150 mathematische Denkspiele, München 1984, p. 20, nr. 15 (Heterogene Quadrate), p. 20, nr. 16 (Antimagie), p. 86, nr. 148 (Höhere Antimagie), pp. 99-100, 178 (Solutions).

Links

• Eric Weisstein's World of Mathematics, Antimagic Square.
• Eric Weisstein's World of Mathematics, Magic Square.
• Eric Weisstein's World of Mathematics, Semimagic Square.

Crossrefs

Cf. A006003, A006052, A050257, A086829, A088020.